Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [8624,2,Mod(1,8624)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8624.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8624, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 8624 = 2^{4} \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8624.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,2,0,-4,0,0,0,0,0,-2,0,-2,0,0,0,-4,0,4,0,0,0,4,0,2,0,2,0, -6,0,8,0,-2,0,0,0,-16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(37)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(68.8629867032\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 154)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 8624.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.41421 q^{3} -0.585786 q^{5} +2.82843 q^{9} -1.00000 q^{11} -3.82843 q^{13} -1.41421 q^{15} +3.65685 q^{17} +0.585786 q^{19} +6.24264 q^{23} -4.65685 q^{25} -0.414214 q^{27} +2.65685 q^{29} +4.00000 q^{31} -2.41421 q^{33} -9.41421 q^{37} -9.24264 q^{39} -5.41421 q^{41} +5.65685 q^{43} -1.65685 q^{45} +10.4853 q^{47} +8.82843 q^{51} +7.89949 q^{53} +0.585786 q^{55} +1.41421 q^{57} +5.58579 q^{59} +11.8284 q^{61} +2.24264 q^{65} -2.75736 q^{67} +15.0711 q^{69} +11.0711 q^{71} -9.41421 q^{73} -11.2426 q^{75} +13.2426 q^{79} -9.48528 q^{81} +12.1421 q^{83} -2.14214 q^{85} +6.41421 q^{87} +12.4853 q^{89} +9.65685 q^{93} -0.343146 q^{95} -3.82843 q^{97} -2.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 4 q^{5} - 2 q^{11} - 2 q^{13} - 4 q^{17} + 4 q^{19} + 4 q^{23} + 2 q^{25} + 2 q^{27} - 6 q^{29} + 8 q^{31} - 2 q^{33} - 16 q^{37} - 10 q^{39} - 8 q^{41} + 8 q^{45} + 4 q^{47} + 12 q^{51}+ \cdots - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 2.41421 1.39385 0.696923 − 0.717146i \(-0.254552\pi\)
0.696923 + 0.717146i \(0.254552\pi\)
\(4\) 0 0
\(5\) −0.585786 −0.261972 −0.130986 − 0.991384i \(-0.541814\pi\)
−0.130986 + 0.991384i \(0.541814\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 2.82843 0.942809
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) −3.82843 −1.06181 −0.530907 − 0.847430i \(-0.678149\pi\)
−0.530907 + 0.847430i \(0.678149\pi\)
\(14\) 0 0
\(15\) −1.41421 −0.365148
\(16\) 0 0
\(17\) 3.65685 0.886917 0.443459 − 0.896295i \(-0.353751\pi\)
0.443459 + 0.896295i \(0.353751\pi\)
\(18\) 0 0
\(19\) 0.585786 0.134389 0.0671943 − 0.997740i \(-0.478595\pi\)
0.0671943 + 0.997740i \(0.478595\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 6.24264 1.30168 0.650840 − 0.759215i \(-0.274417\pi\)
0.650840 + 0.759215i \(0.274417\pi\)
\(24\) 0 0
\(25\) −4.65685 −0.931371
\(26\) 0 0
\(27\) −0.414214 −0.0797154
\(28\) 0 0
\(29\) 2.65685 0.493365 0.246683 − 0.969096i \(-0.420659\pi\)
0.246683 + 0.969096i \(0.420659\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 − 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 0 0
\(33\) −2.41421 −0.420261
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −9.41421 −1.54769 −0.773844 − 0.633377i \(-0.781668\pi\)
−0.773844 + 0.633377i \(0.781668\pi\)
\(38\) 0 0
\(39\) −9.24264 −1.48001
\(40\) 0 0
\(41\) −5.41421 −0.845558 −0.422779 − 0.906233i \(-0.638945\pi\)
−0.422779 + 0.906233i \(0.638945\pi\)
\(42\) 0 0
\(43\) 5.65685 0.862662 0.431331 − 0.902194i \(-0.358044\pi\)
0.431331 + 0.902194i \(0.358044\pi\)
\(44\) 0 0
\(45\) −1.65685 −0.246989
\(46\) 0 0
\(47\) 10.4853 1.52944 0.764718 − 0.644365i \(-0.222878\pi\)
0.764718 + 0.644365i \(0.222878\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 8.82843 1.23623
\(52\) 0 0
\(53\) 7.89949 1.08508 0.542540 − 0.840030i \(-0.317463\pi\)
0.542540 + 0.840030i \(0.317463\pi\)
\(54\) 0 0
\(55\) 0.585786 0.0789874
\(56\) 0 0
\(57\) 1.41421 0.187317
\(58\) 0 0
\(59\) 5.58579 0.727207 0.363604 − 0.931554i \(-0.381546\pi\)
0.363604 + 0.931554i \(0.381546\pi\)
\(60\) 0 0
\(61\) 11.8284 1.51447 0.757237 − 0.653140i \(-0.226549\pi\)
0.757237 + 0.653140i \(0.226549\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 2.24264 0.278165
\(66\) 0 0
\(67\) −2.75736 −0.336865 −0.168433 − 0.985713i \(-0.553871\pi\)
−0.168433 + 0.985713i \(0.553871\pi\)
\(68\) 0 0
\(69\) 15.0711 1.81434
\(70\) 0 0
\(71\) 11.0711 1.31389 0.656947 − 0.753937i \(-0.271848\pi\)
0.656947 + 0.753937i \(0.271848\pi\)
\(72\) 0 0
\(73\) −9.41421 −1.10185 −0.550925 − 0.834555i \(-0.685725\pi\)
−0.550925 + 0.834555i \(0.685725\pi\)
\(74\) 0 0
\(75\) −11.2426 −1.29819
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 13.2426 1.48991 0.744957 − 0.667113i \(-0.232470\pi\)
0.744957 + 0.667113i \(0.232470\pi\)
\(80\) 0 0
\(81\) −9.48528 −1.05392
\(82\) 0 0
\(83\) 12.1421 1.33277 0.666386 − 0.745607i \(-0.267840\pi\)
0.666386 + 0.745607i \(0.267840\pi\)
\(84\) 0 0
\(85\) −2.14214 −0.232347
\(86\) 0 0
\(87\) 6.41421 0.687676
\(88\) 0 0
\(89\) 12.4853 1.32344 0.661719 − 0.749752i \(-0.269827\pi\)
0.661719 + 0.749752i \(0.269827\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 9.65685 1.00137
\(94\) 0 0
\(95\) −0.343146 −0.0352060
\(96\) 0 0
\(97\) −3.82843 −0.388718 −0.194359 − 0.980930i \(-0.562263\pi\)
−0.194359 + 0.980930i \(0.562263\pi\)
\(98\) 0 0
\(99\) −2.82843 −0.284268
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 8624.2.a.cc.1.2 2
4.3 odd 2 1078.2.a.t.1.1 2
7.2 even 3 1232.2.q.f.529.1 4
7.4 even 3 1232.2.q.f.177.1 4
7.6 odd 2 8624.2.a.bh.1.1 2
12.11 even 2 9702.2.a.cx.1.1 2
28.3 even 6 1078.2.e.m.177.1 4
28.11 odd 6 154.2.e.e.23.2 ✓ 4
28.19 even 6 1078.2.e.m.67.1 4
28.23 odd 6 154.2.e.e.67.2 yes 4
28.27 even 2 1078.2.a.x.1.2 2
84.11 even 6 1386.2.k.t.793.2 4
84.23 even 6 1386.2.k.t.991.2 4
84.83 odd 2 9702.2.a.ch.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
154.2.e.e.23.2 ✓ 4 28.11 odd 6
154.2.e.e.67.2 yes 4 28.23 odd 6
1078.2.a.t.1.1 2 4.3 odd 2
1078.2.a.x.1.2 2 28.27 even 2
1078.2.e.m.67.1 4 28.19 even 6
1078.2.e.m.177.1 4 28.3 even 6
1232.2.q.f.177.1 4 7.4 even 3
1232.2.q.f.529.1 4 7.2 even 3
1386.2.k.t.793.2 4 84.11 even 6
1386.2.k.t.991.2 4 84.23 even 6
8624.2.a.bh.1.1 2 7.6 odd 2
8624.2.a.cc.1.2 2 1.1 even 1 trivial
9702.2.a.ch.1.2 2 84.83 odd 2
9702.2.a.cx.1.1 2 12.11 even 2